| L(s) = 1 | − 6.38·2-s + 8.83·4-s + 38.7·5-s + 148.·8-s − 247.·10-s + 576.·11-s − 391.·13-s − 1.22e3·16-s − 1.32e3·17-s − 942.·19-s + 341.·20-s − 3.68e3·22-s + 1.63e3·23-s − 1.62e3·25-s + 2.50e3·26-s + 1.46e3·29-s + 3.91e3·31-s + 3.11e3·32-s + 8.49e3·34-s − 1.63e4·37-s + 6.02e3·38-s + 5.73e3·40-s − 1.31e4·41-s + 1.47e4·43-s + 5.08e3·44-s − 1.04e4·46-s − 6.81e3·47-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.275·4-s + 0.692·5-s + 0.817·8-s − 0.782·10-s + 1.43·11-s − 0.642·13-s − 1.19·16-s − 1.11·17-s − 0.598·19-s + 0.191·20-s − 1.62·22-s + 0.643·23-s − 0.520·25-s + 0.725·26-s + 0.323·29-s + 0.731·31-s + 0.537·32-s + 1.26·34-s − 1.95·37-s + 0.676·38-s + 0.566·40-s − 1.21·41-s + 1.21·43-s + 0.396·44-s − 0.726·46-s − 0.449·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 6.38T + 32T^{2} \) |
| 5 | \( 1 - 38.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 576.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 391.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 942.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.63e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.63e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.31e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.47e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.81e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.01e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.56e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.12e3T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.712000892620380072860334888702, −9.012427557239995462750680318216, −8.399402264299868026734800389322, −7.07295666054133509186864911561, −6.47697299208237884606306989848, −5.03968112962672072890145470059, −3.99043836664774357386062073019, −2.24669176330310343385865794890, −1.31064231601552648464275750368, 0,
1.31064231601552648464275750368, 2.24669176330310343385865794890, 3.99043836664774357386062073019, 5.03968112962672072890145470059, 6.47697299208237884606306989848, 7.07295666054133509186864911561, 8.399402264299868026734800389322, 9.012427557239995462750680318216, 9.712000892620380072860334888702
